Average Error: 39.0 → 6.2
Time: 21.3s
Precision: 64
Internal precision: 2688
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b \le -7.775815916098481 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{\frac{4}{\frac{2}{c}}}{\frac{c}{b} \cdot a - b}}{2}\\ \mathbf{if}\;b \le 2.4861451900721143 \cdot 10^{+28}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original39.0
Comparison26.8
Herbie6.2
\[ \begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes.

  2. if b < -7.775815916098481e-103

    1. Initial program 58.7

      \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-- 58.7

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    4. Applied simplify 35.5

      \[\leadsto \frac{\frac{\color{blue}{a \cdot \left(4 \cdot c\right)}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied div-inv 35.5

      \[\leadsto \frac{\color{blue}{\left(a \cdot \left(4 \cdot c\right)\right) \cdot \frac{1}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    7. Applied times-frac 34.9

      \[\leadsto \color{blue}{\frac{a \cdot \left(4 \cdot c\right)}{2} \cdot \frac{\frac{1}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}}\]

    8. Applied taylor 18.1

      \[\leadsto \frac{a \cdot \left(4 \cdot c\right)}{2} \cdot \frac{\frac{1}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{a}\]

    9. Taylor expanded around -inf 18.1

      \[\leadsto \frac{a \cdot \left(4 \cdot c\right)}{2} \cdot \frac{\frac{1}{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}}{a}\]

    10. Applied simplify 4.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{4}{\frac{2}{c}}}{\frac{c}{b} \cdot a - b}}{2}}\]

    if -7.775815916098481e-103 < b < 2.4861451900721143e+28

    1. Initial program 12.4

      \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    if 2.4861451900721143e+28 < b

    1. Initial program 39.1

      \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Applied taylor 0

      \[\leadsto -1 \cdot \frac{b}{a}\]

    3. Taylor expanded around inf 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]

    4. Applied simplify 0

      \[\leadsto \color{blue}{\frac{-b}{a}}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 21.3s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064651971 495577305 2200811460 13024471 864198081 231948279)'
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))